Alex flips a fair coin 20 times. Bert spins a fair coin 21
times. Bert wins if he gets more heads than Alex, else Alex wins.
Note that Alex wins if there is a tie. What is the probability that
Bert wins?

Gururaj,
I agree with you...
I made up a full probability distribution table and the contrasted all possible outcomes.
The probability of any one outcome for Alex: 0.5^20
multiplied by the number of combinations of that many heads out of the 20 spins: (20 C x).
As for Bert: 0.5^21 and (21 C x).
It was then a matter of contrasting these against each other and summing for a total probablility of a win for each player.
The final answer... they both have a 0.5 chance of winning.